Mathematics for the interested outsider

Lie Algebras Revisited

Well it’s been quite a while, but I think I can carve out the time to move forwards again. I was all set to start with Lie algebras today, only to find that I’ve already defined them over a year ago. So let’s pick up with a recap: a Lie algebra is a module — usually a vector space over a field — called and give it a bilinear operation which we write as . We often require such operations to be associative, but this time we impose the following two conditions:

Now, as long as we’re not working in a field where — and usually we’re not — we can use bilinearity to rewrite the first condition:

so . This antisymmetry always holds, but we can only go the other way if the character of is not , as stated above.

The second condition is called the “Jacobi identity”, and antisymmetry allows us to rewrite it as:

That is, bilinearity says that we have a linear mapping that sends an element to a linear endomorphism in . And the Jacobi identity says that this actually lands in the subspace of “derivations” — those which satisfy something like the Leibniz rule for derivatives. To see what I mean, compare to the product rule:

where takes the place of , takes the place of , and takes the place of . And the operations are changed around. But you should see the similarity.

Lie algebras obviously form a category whose morphisms are called Lie algebra homomorphisms. Just as we might expect, such a homomorphism is a linear map that preserves the bracket:

We can obviously define subalgebras and quotient algebras. Subalgebras are a bit more obvious than quotient algebras, though, being just subspaces that are closed under the bracket. Quotient algebras are more commonly called “homomorphic images” in the literature, and we’ll talk more about them later.

We will take as a general assumption that our Lie algebras are finite-dimensional, though infinite-dimensional ones absolutely exist and are very interesting.

Yes, they’re submodules in the more general situation of a Lie algebra over a ring, but we’re just going to be looking at Lie algebras over fields — and usually characteristic zero and almost always algebraically closed, to boot.

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This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.